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Fabrice Mouhartem 2018-02-28 18:05:08 +01:00
parent d74450ebac
commit 4128e63832
2 changed files with 6 additions and 6 deletions
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4
main.tex
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@ -39,7 +39,6 @@
\usepackage{amsthm} % For theorem style \usepackage{amsthm} % For theorem style
\usepackage{thmtools} \usepackage{thmtools}
\usepackage{thm-restate} \usepackage{thm-restate}
\usepackage[capitalise]{cleveref}
\usepackage{comment} \usepackage{comment}
\usepackage{tikz} \usepackage{tikz}
\usetikzlibrary{positioning,patterns,shapes} \usetikzlibrary{positioning,patterns,shapes}
@ -49,6 +48,9 @@
\declaretheorem[sibling=theorem]{lemma} \declaretheorem[sibling=theorem]{lemma}
\declaretheorem[numberwithin=chapter,style=definition]{definition} \declaretheorem[numberwithin=chapter,style=definition]{definition}
% References
\usepackage[capitalise]{cleveref}
\usepackage{pdfpages} \usepackage{pdfpages}
\usepackage{xspace} \usepackage{xspace}

8
sec-lattices.tex
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@ -62,7 +62,7 @@ In order to define the $\SIVP$ problem and assumption, let us first define the s
where $\mathcal B(\mathbf c, r)$ denotes the ball of radius $r$ centered in $\mathbf c$. where $\mathcal B(\mathbf c, r)$ denotes the ball of radius $r$ centered in $\mathbf c$.
\end{definition} \end{definition}
Which leads us to the $\SIVP$ problem, which is finding a set of sufficiently short linearly independent vectors given a lattice basis. This leads us to the $\SIVP$ problem, which is finding a set of sufficiently short linearly independent vectors given a lattice basis.
\begin{definition}[$\SIVP$] \label{de:sivp} \begin{definition}[$\SIVP$] \label{de:sivp}
For a dimension $n$ lattice described by a basis $\mathbf B \in \RR^{n \times m}$, and a parameter $\gamma > 0$, the shortest independent vectors problem is to find $n$ linearly independent vectors $v_1, \ldots, v_n$ such that $\| v_1 \| \leq \| v_2 \| \leq \ldots \leq \| v_n \|$ and $\|v_n\| \leq \gamma \cdot \lambda_n(\mathbf B)$. For a dimension $n$ lattice described by a basis $\mathbf B \in \RR^{n \times m}$, and a parameter $\gamma > 0$, the shortest independent vectors problem is to find $n$ linearly independent vectors $v_1, \ldots, v_n$ such that $\| v_1 \| \leq \| v_2 \| \leq \ldots \leq \| v_n \|$ and $\|v_n\| \leq \gamma \cdot \lambda_n(\mathbf B)$.
@ -73,10 +73,8 @@ In other words, it means that no polynomial time algorithms can solve those prob
%As explained before, we will rely on the assumption that both algorithmic problems below are hard. Meaning that no (probabilistic) polynomial time algorithms can solve them with non-negligible probability and non-negligible advantage, respectively. %As explained before, we will rely on the assumption that both algorithmic problems below are hard. Meaning that no (probabilistic) polynomial time algorithms can solve them with non-negligible probability and non-negligible advantage, respectively.
\begin{definition}[The $\SIS$ problem] \label{de:sis} \index{Lattices!Short Integer Solution} \begin{definition}[The $\SIS$ problem] \label{de:sis} \index{Lattices!Short Integer Solution}
Let~$m,q,\beta$ be functions of~$n \in \mathbb{N}$. The Short Integer Let~$m,q,\beta$ be functions of~$n \in \mathbb{N}$.
Solution problem $\SIS_{n,m,q,\beta}$ is, given~$\mathbf{A} \sample The Short Integer Solution problem $\SIS_{n,m,q,\beta}$ is, given~$\mathbf{A} \sample U(\Zq^{n \times m})$, find~$\mathbf{x} \in \Lambda_q^{\perp}(\mathbf{A})$ with~$0 < \|\mathbf{x}\| \leq \beta$.
U(\Zq^{n \times m})$, find~$\mathbf{x} \in \Lambda_q^{\perp}(\mathbf{A})$
with~$0 < \|\mathbf{x}\| \leq \beta$.
\end{definition} \end{definition}
If~$q \geq \sqrt{n} \beta$ and~$m,\beta \leq \mathsf{poly}(n)$, then $\SIS_{n,m,q,\beta}$ is at least as hard as If~$q \geq \sqrt{n} \beta$ and~$m,\beta \leq \mathsf{poly}(n)$, then $\SIS_{n,m,q,\beta}$ is at least as hard as